Prior to explaining the invention, it will be helpful to first understand the prior art of both the Kalman Filter (KF) and the Fast Kalman Filter (FKF.TM.) for calibrating a sensor system (WO 90/13794). The underlying Markov process is described by the equations from (1) to (3). The first equation tells how a measurement vector y.sub.t depends on the state vector s.sub.t at timepoint t, (t=0,1,2 . . . ). This is the linearized Measurement (or observation) equation: EQU y.sub.t =H.sub.t s.sub.t +e.sub.t ( 1)
The design matrix H.sub.t is typically composed of the partial derivatives of the actual Measurement equations. The second equation describes the time evolution of e.g. a weather balloon flight and is the System (or state) equation: EQU s.sub.t =s.sub.t-1 +u.sub.t-1 +a.sub.t ( 2)
(or, s.sub.t =As.sub.t-1 +Bu.sub.t-1 +a.sub.t more generally) PA1 cc.sub.t the estimated calibration parameters. PA1 b.sub.t,k all other state parameters in the time and/or space volume PA1 A state transition matrix (bock-diagonal) at time t PA1 B matrix (bock-diagonal) for state-independent effects u.sub.t at time t. PA1 Augmented model for a space volume case: see also equations (15) and (16), e.g. for the data of an entire windtracking experiment with K consecutive balloon positions: ##STR1## Augmented Model for a moving time volume (e.g. for "whitening" an observed "innovations" sequence of residuals e.sub.t over a moving sample of length L): ##STR2##
which tells how the balloon position is composed of its previous position s.sub.t-1 as well as of increments u.sub.t-1 and a.sub.t These increments are typically caused by a known uniform motion and an unknown random acceleration, respectively.
The measurement errors, the acceleration term and the previous position usually are mutually uncorrelated and are briefly described here by the following covariance matrices: EQU R.sub.e.sbsb.t =Cov(e.sub.t)=E(e.sub.t e.sub.t ') EQU R.sub.a.sbsb.t =Cov(a.sub.t)=E(a.sub.t a.sub.t ')
and EQU P.sub.t-1 =Cov(s.sub.t-1)=E{(s.sub.t-1 -s.sub.t-1)(s.sub.t-1 -s.sub.t-1)'}(3)
The Kalman forward recursion formulae give us the best linear unbiased estimates of the present state EQU s.sub.t =s.sub.t-1 +u.sub.t-1 +K.sub.t {y.sub.t -H.sub.t (s.sub.t-1 +u.sub.t-1)} (4)
and its covariance matrix EQU P.sub.t =Cov(s.sub.t)=P.sub.t-1 -K.sub.t H'.sub.t P.sub.t-1( 5)
where the Kalman gain matrix K.sub.t is defined by EQU K.sub.t =(P.sub.t-1 +R.sub.a.sbsb.t)H'.sub.t {H.sub.t (P.sub.t-1 R.sub.a.sbsb.t)H'.sub.t +R.sub.e.sbsb.t }.sup.-1 ( 6)
Let us now partition the estimated state vector s.sub.t and its covariance matrix P.sub.t as follows: ##EQU1## where b.sub.t tells us the estimated balloon position; and,
The respective partitioning of the other quantities will then be as follows: ##EQU2##
The recursion formulae from (4) to (6) gives us now a filtered (based on updated calibration parameters) position vector EQU b.sub.t =b.sub.t-1 +u.sub.b.sbsb.t-1 +K.sub.b.sbsb.t {y.sub.t -H.sub.t (s.sub.t-1 +u.sub.t-1)} (9)
and the updated calibration parameter vector EQU c.sub.t =c.sub.t-1 +u.sub.c.sbsb.t-1 +K.sub.c.sbsb.t {y.sub.t -H.sub.t (st-1+u.sub.t-1)} (10)
The Kalman gain matrices are respectively ##EQU3##
The following modified form of the general State equation is introduced EQU As.sub.t-1 +Bu.sub.t-1 =Is.sub.t +A(s.sub.t-1 -s.sub.t-1)-a.sub.t( 12)
where s represents an estimated value of a state vector s. Combine it with the Measurement equation (1) in order to obtain so-called Augmented Model: ##EQU4## The state parameters can now be computed by using the well-known solution of a Regression Analysis problem given below. Use it for Updating: EQU s.sub.t =(Z'.sub.t V.sub.t.sup.-1 Z.sub.t).sup.-1 Z'.sub.t V.sub.t.sup.-1 z.sub.t ( 14)
The result is algebraically equivalent to use of the Kalman Recursions but not numerically. For the balloon tracking problem with a large number sensors with slipping calibration the matrix to be inverted in equations (6) or (11) is larger than that in formula (14).
The initialization of the large Fast Kalman Filter (FKF.TM.) for solving the calibration problem of the balloon tracking sensors is done by Lange's High-pass Filter. It exploits an analytical sparse-matrix inversion formula (Lange, 1988a) for solving regression models with the following so-called Canonical Block-angular matrix structure: ##EQU5## This is a matrix representation of the Measurement equation of an entire windfinding intercomparison experiment or one balloon flight. The vectors b.sub.1,b.sub.2, . . . ,b.sub.K typically refer to consecutive position coordinates of a weather balloon but may also contain those calibration parameters that have a significant time or space variation. The vector c refers to the other calibration parameters that are constant over the sampling period.
The Regression Analytical approach of the Fast Kalman Filtering (FKF.TM.) for updating the state parameters including the calibration drifts in particular, is based on the same block-angular matrix structure as in equation (15). The optimal estimates ( ) of b.sub.1,b.sub.2, . . . b.sub.K and c are obtained by making the following logical insertions into formula (15) for each timepoint t, t=1,2, . . . : ##EQU6## These insertions concluded the specification of the Fast Kalman Filter (FKF.TM.) algorithm for calibrating the upper-air wind tracking system. Another application would be the Global Observing System of the World Weather Watch. Here, the vector y.sub.k contains various observed inconsistencies and systematic errors of weather reports (e.g. mean day-night differences of pressure values which should be about zero) from a radiosonde system k or from a homogeneous cluster k of radiosonde stations of a country (Lange, 1988a/b). The calibration drift vector b.sub.k will then tell us what is wrong and to what extent. The calibration drift vector c refers to errors of a global nature or which are more or less common to all observing systems (e.g. biases in satellite radiances and in their vertical weighting functions or some atmospheric tide effects).
For all these large multiple sensor systems their design matrices H typically are sparse. Thus, one can usually perform in one way or another the following sort of ##EQU7## where c.sub.t typically represents calibration parameters at time t
Consequently, two (or three) types of gigantic Regression Analysis problems EQU Z.sub.t =Z.sub.t S.sub.t +e.sub.t ( 18)
were faced as follows:
Please observe that the matrix formula may take a "nested" Block-Angular structure. Fast semi-analytical solutions based on EQU Updating: S.sub.t ={Z'.sub.t V.sub.t.sup.-1 Z.sub.t }.sup.- Z'.sub.t V.sub.t.sup.-1 Z.sub.t ( 19)
for all these three cases were published in PCT/FI90/00122 (Lange, 1990), WIPO, Geneva, Switzerland.
The Fast Kalman Filter (FKF.TM.) formulae for the recursion step at any timepoint t were as follows: ##EQU8## where, for l=0,1,2, . . . , L-1, ##EQU9## and, i.e. for l=L, EQU R.sub.t-L =V.sub.t-L.sup.-1 EQU V.sub.t-L =Cov{A(C.sub.t-1 -C.sub.t-1)-a.sub.c.sbsb.t } EQU y.sub.t-L =AC.sub.t-1 +Bu.sub.c.sbsb.t-1 EQU G.sub.t-L =I.
A major R & D project was initiated in 1988 which led to the start of cooperation between ECMWF and Meteo-France for the coding of a dynamical atmospheric model, an optimal interpolation, a variational data assimilation and a Kalman Filter (FK), all in the same framework. The project is called IFS (Integrated Forecasting System), see Jean-Noel Thepaut and Philippe Courtier ( 1991): "Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model", Quarterly Journal of the Royal Meteorological Society, Volume 117, pp. 1225-1254.
Similar Kalman Filter (FK) studies have recently been reported also by Roger Daley (1992): "The Lagged Innovation Covariance: A Performance Diagnostic for Atmospheric Data Assimilation", Monthly Weather Review of the American Meteorological Society, Vol. 120, pp. 178-196, and Stephen E. Cohn and David F. Parrish (1991): "The Behavior of Forecast Error Covariances for a Kalman Filter in Two Dimensions", Monthly Weather Review of the American Meteorological Society, Vol. 119, pp. 1757-1785. Unfortunately, the ideal Kalman Filter systems described in the above reports have been out of reach at the present time. Dr. T. Gal-Chen of School of Meteorology, University of Oklahoma, reported in May 1988: "There is hope that the developments of massively parallel super computers (e.g., 1000 desktop CRAYs working in tandem) could result in algorithms much closer to optimal . . . ", see "Report of the Critical Review Panel--Lower Tropospheric Profiling Symposium: Needs and Technologies", Bulletin of the American Meteorological Society, Vol. 71, No. 5, May 1990, page 684.
There exists a need for exploiting the principles of the Fast Kalman Filtering (FKF.TM.) method for a broad technical field (broader than just calibrating a sensor system in some narrow sense of word "calibration") with equal or better computational speed, reliability, accuracy, and cost benefits than other Kalman Filtering methods can do.